Topological Crystals

Abstract

Sunada's work on crystallography emphasizes the role of the 'maximal abelian cover' of a graph X. This is a covering space of X for which the group of deck transformations is the first homology group H₁(X,Z). An embedding of the maximal abelian cover in a vector space can serve as the pattern for a crystal: atoms are located at the vertices, while bonds lie along the edges. We prove that for any connected graph X without bridges, there is a canonical embedding of the maximal abelian cover of X into the vector space H₁(X,R), called a 'topological crystal'.  We prove that any symmetry of a graph lifts to a symmetry of its topological crystal.  The key technical tools are a way of decomposing the 1-chain coming from a path in X into manageable pieces, and the work of Bacher, de la Harpe and Nagnibeda on integral cycles and integral cuts.